ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION
JIAWANG NIE AND KRISTIAN RANESTAD
Abstract. Consider the polynomial optimization problem whose objective and constraints
are all described by multivariate polynomials. Under some genericity assumptions, we prove
that the optimality conditions always hold on optimizers, and the coordinates of optimizers
are algebraic functions of the coefficients of the input polynomials. We also give a general
formula for the algebraic degree of the optimal coordinates. The derivation of the algebraic
degree is equivalent to counting the number of all complex critical points. As special cases,
we obtain the algebraic degrees of quadratically constrained quadratic programming (QCQP),
second order cone programming (SOCP) and p-th order cone programming (POCP), in analogy
to the algebraic degree of semidefinite programming [9].
1. Introduction
Consider the optimization problem


minn
f0 (x)
 x∈R
(1.1)
s.t.
fi (x) = 0, i = 1, . . . , me


fi (x) ≥ 0, i = me + 1, . . . , m
where the fi are multivariate polynomial functions in R[x] (the ring of polynomials in x =
(x1 , . . . , xn ) with real coefficients). The recent interest on solving polynomial optimization problems [7, 8, 11, 12] by using semidefinite relaxations or other algebraic methods motivates this
study of the algebraic properties of the polynomial optimization problem (1.1). A fundamental
question regarding (1.1) is how the optimal solutions depend on the input polynomials fi . When
the optimality condition holds and the critical equations of (1.1) have finitely many complex
solutions, the optimal solutions are algebraic functions of the coefficients of the polynomials
fi , in particular, the coordinates of optimal solutions are roots of some univariate polynomials
whose coefficients are functions of the input data. An interesting and important problem in
optimization theory is to find the degrees of these algebraic functions as functions of the degrees
of the fi , which amounts to computing the number of complex solutions to the critical equations
of (1.1). We begin our discussion with some special cases.
The simplest case of (1.1) is linear programming (LP), when all the polynomials fi have degree
one. In this case, the problem (1.1) has the form (after removing the linear equality constraints)
(
minn
cT x
x∈R
(1.2)
s.t.
Ax ≥ b
where c, A, b are matrices or vectors of appropriate dimensions. The feasible set of (1.2) is now
a polyhedron described by some linear inequalities. As is well-known, if the polyhedron has a
vertex and (1.2) has an optimal solution, then an optimizer x∗ of (1.2) occurs at a vertex. So x∗
Key words and phrases. algebraic degree, polynomial optimization, optimality condition, quadratically constrained quadratic programming (QCQP), p-th order cone programming, second order cone programming (SOCP),
variety .
1
2
JIAWANG NIE AND KRISTIAN RANESTAD
can be determined by the linear system consisting of the active constraints. When the objective
cT x is changing, the optimal solution might move from one vertex to another vertex. So the
optimal solution is a piecewise linear fractional function of the input data (c, A, b). When the
c, A, b are all rational, an optimal solution must also be rational, and hence its algebraic degree
is one.
A more general convex optimization which is a proper generalization of linear programming
is semidefinite programming (SDP). It has the standard form


minn
cT x
 x∈R
n
(1.3)
P

A0 + xi Ai º 0
 s.t.
i=1
where c is a constant vector and the Ai are constant symmetric matrices. The inequality X º 0
means the matrix X is positive semidefinite. Recently, Nie et al. [9] studied the algebraic
properties of semidefinite programming. When c and Ai are generic, the optimal solution x∗ of
(1.3) is shown in [9] to be a piecewise algebraic function of c and Ai . Of course, the constraint of
(1.3) can be replaced by the nonnegativity of all the principle minors of the constraint matrix,
and hence (1.3) becomes a special case of (1.1). However, the problem (1.3) has very special
nice properties, e.g., it is a convex program and the constraint matrix is linear with respect to
x. Interestingly, if c and Ai are generic, the degree of each piece of this algebraic function only
depends on the rank of the constraint matrix at the optimal solution. A formula for this degree
is given in [13, Theorem 1.1].
Another optimization problem frequently used in statistics and biology is the Maximum Likelihood Estimation (MLE) problem. It has the standard form
(1.4)
max p1 (x)u1 p2 (x)u2 · · · pm (x)um
x∈Θ
P
where Θ is an open subset of Rn , the pi are polynomials such that i pi = 1, and the ui are
given positive integers. The optimizer x∗ is an algebraic function of (u1 , . . . , un ). This problem
has recently been studied and a formula for the degree of this algebraic function has been found
(cf. [1, 6]).
In this paper we consider the general optimization problem (1.1) when the polynomials
f0 , f1 , . . . , fm define a complete intersection, i.e., their common set of zeros has codimension
m + 1 (see the appendix for the definition of complete intersection). We show that an optimal
solution is an algebraic function of the input data. We call the degree of this algebraic function
the algebraic degree of the polynomial optimization problem (1.1). Equivalently, the algebraic
degree equals the number of complex solutions to the critical equations of (1.1) when this number
is finite. Under some genericity assumptions, we give in this paper a formula for the algebraic
degree of (1.1).
Throughout this paper, the words “generic” and “genericity” are frequently used. We shall
use them as conditions on the input data for some property to hold, and they shall mean for all
but a set of Lebesgue measure zero in the space of data.
The algebraic degree of polynomial optimization (1.1) addresses the computational complexity
at a fundamental level. To solve (1.1) exactly essentially reduces to solving some univariate
polynomial equations whose degrees are the algebraic degree of (1.1). As we will see, the
algebraic degree grows rapidly with the degrees of the fi .
The paper is organized as follows: In Section 2 we derive a general formula for the algebraic
degree, and in Section 3 we give the formulae of the algebraic degrees for special cases like
quadratically constrained quadratic programming, second order cone programming, and p-th
ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION
3
order cone programming. The paper is concluded with an appendix which introduces some
basic concepts and facts in algebraic geometry that is necessary for this paper.
2. A general formula for the algebraic degree
In this section we will derive a formula for the algebraic degree of the polynomial optimization
problem (1.1) when the polynomials define a complete intersection. Suppose the polynomial fi
has degree di . Let x∗ be a local or global optimal solution of (1.1). At first, we assume that
the polynomials are general and all the inequality constraints are active, i.e., me = m. When
m = n, by Corollary A.2, the feasible set of (1.1) is finite and hence the algebraic degree is equal
to the product of the degrees of the polynomials, d1 d2 · · · dm . So, from now on, assume m < n.
If the variety
V = {x ∈ Cn : f1 (x) = · · · = fm (x) = 0}
is smooth at x∗ , i.e., the gradient vectors
∇f1 (x∗ ), ∇f2 (x∗ ), . . . , ∇fm (x∗ )
are linearly independent, then the Karush-Kuhn-Tucker (KKT) condition holds at x∗ (Chapter 12 in [10]). In fact,

m
 ∇f (x∗ ) + P
λ∗i ∇fi (x∗ ) = 0
0
(2.1)
i=1

f1 (x∗ ) = · · · = fm (x∗ ) = 0
where λ∗1 , . . . , λ∗m are Lagrange multipliers for the constraints f1 (x) = 0, . . . , fm (x) = 0. Thus
the optimal solution x∗ and the Lagrange multipliers λ∗ = (λ∗1 , . . . , λ∗m ) are determined by
the polynomial system (2.1). The set of complex solutions to (2.1) forms the locus of complex
critical points of (1.1). If the system (2.1) is zero-dimensional, then, by elimination theory (cf.[2,
Chapter 3]), the coordinates of x∗ are algebraic functions of the coefficients of the polynomials
fi . Each coordinate x∗i can be determined by some univariate polynomial equation like
(x∗i )δi + a1 (x∗i )δi −1 + · · · + aδi −1 x∗i + aδi = 0
where aj are rational functions of the coefficients of the fi . Interestingly, when f1 , f2 , . . . , fm are
generic, the KKT condition always holds at any optimal solution, and the degrees δi are equal
to each other. This common degree counts the number of complex solutions to (2.1), i.e., the
cardinality of the complex critical locus of (1.1) or, by definition, the algebraic degree of the
polynomial optimization (1.1). We will derive a general formula for this degree.
We turn to complex projective spaces, where the above question may be answered as a problem
in intersection theory. For this we translate the optimization problem to a relevant intersection
problem. Let Pn be the n-dimensional complex projective space. A point x̃ ∈ Pn has homogeneous coordinates [x0 , x1 , . . . , xn ] unique up to multiplication by a common nonzero scalar. A
variety in Pn is a set of points x̃ that satisfy a collection of homogeneous polynomial equations
in [x0 , x1 , . . . , xn ]. Let f˜i , defined by f˜i (x̃) = x0di fi (x1 /x0 , . . . , xn /x0 ), be the homogenization of
fi . Define U to be the projective variety
U = {x̃ ∈ Pn : f˜1 (x̃) = f˜2 (x̃) = · · · = f˜m (x̃) = 0}
in Pn . See the appendix for more about projective spaces and projective varieties. Next, let
h
iT
˜ f˜i = ∂ f˜i . . . ∂ f˜i
∇
∂x0
∂xn
4
JIAWANG NIE AND KRISTIAN RANESTAD
be the gradient vector with respect to the homogeneous coordinates. Notice that ( ∂x∂ j f˜i (x̃) =
xd0i −1 ∂x∂ j fi (x1 /x0 , . . . , xn /x0 )), so the homogenization of ∇fi coincides with the last n coordi˜ f˜i .
nates ∇f˜i in ∇
In this homogeneous setting, the optimality condition for problem (1.1) with m = me is
½
¾
f˜0 (x̃)£ − µx0d0 = f˜1 (x̃) = · · · = f˜m (x̃) = 0
n
¤
(2.2)
(x, µ) ∈ C × C :
˜ f˜0 (x̃) + µxd0 ), ∇(
˜ f˜1 (x̃)), . . . , ∇(
˜ f˜m (x̃)) ≤ m
rank ∇(
0
where µ ∈ R is the critical value. Let x̃∗ ∈ {x0 6= 0} be a critical point, i.e., a solution to (2.2).
We may eliminate µ by asking that the matrix
·
¸
f˜0 (x̃∗ ) f˜1 (x̃∗ ) . . . f˜m (x̃∗ )
xd00
0
...
0
have rank at most one and that the matrix

∂ ˜
f0 (x̃∗ ) ∂x∂ 0 f˜1 (x̃∗ ) . . .
 ∂x∂ 0 ˜ ∗
 ∂x1 f0 (x̃ ) ∂x∂ 1 f˜1 (x̃∗ ) . . .

..
..
..

.
.
.

∂ ˜
∂ ˜
∗
∗
∂xn f0 (x̃ ) ∂xn f1 (x̃ ) . . .

(d0 − 1)xd00

0


..
..

.
.

∂ ˜
∗
0
∂xn fm (x̃ )
∂ ˜
∗
∂x0 fm (x̃ )
∂ ˜
∗
∂x1 fm (x̃ )
have rank at most m + 1. Since x0 6= 0, the first condition implies that
x̃∗ ∈ U = {f˜1 (x̃) = · · · = f˜m (x̃) = 0}.
Similarly, the rank of the second matrix equals m + 1 only if the submatrix
 ∂ ˜

∂ ˜
∂ ˜
∂x1 f0 (x̃)
∂x1 f1 (x̃) . . . ∂x1 fm (x̃)


..
..
..
..
M =

.
.
.
.
∂ ˜
∂ ˜
∂ ˜
f0 (x̃)
f1 (x̃) . . .
fm (x̃)
∂xn
∂xn
∂xn
has rank m. Therefore we define W to be the projective variety in Pn
W = {x̃ ∈ Pn : all the (m + 1) × (m + 1) minors of M vanish } ,
i.e., the locus of points where the rank of [∇f˜0 (x̃), . . . , ∇f˜m (x̃)] is less than or equal to m.
Proposition 2.1. Consider the polynomial optimization problem (1.1), and assume that m =
me , i.e., that all constraints are active. If the polynomials f1 , . . . , fm are generic, then we have:
(i) The affine variety V = {x ∈ Cn : f1 (x) = · · · = fm (x) = 0} is smooth;
(ii) The KKT condition holds at any optimal solution x∗ ;
(iii) If f0 is also generic, the affine variety
)
(
m
X
λi ∇fi (x) = 0
(2.3)
K = x ∈ V : ∃ λ1 , . . . , λm such that ∇f0 (x) +
i=1
defined by the KKT system (2.1) is finite.
Proof. (i) When the polynomials f1 , . . . , fm are generic, then their homogenizations f˜1 , . . . , f˜m
are generic, so by Corollary A.2, they define a smooth complete intersection variety U of codimension m. In particular the affine subvariety V = U ∩ {x0 6= 0} is smooth. Therefore the
ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION
Jacobian matrix

∂ ˜
∂x0 f1 (x̃)
 ∂ ˜
 ∂x1 f1 (x̃)



..
.
∂ ˜
∂xn f1 (x̃)
∂ ˜
∂x0 f2 (x̃)
∂ ˜
∂x1 f2 (x̃)
...
...
..
.
..
.
∂ ˜
∂xn f2 (x̃) . . .
5

∂ ˜
∂x0 fm (x̃)

∂ ˜
∂x1 fm (x̃) 

..

.

∂ ˜
∂xn fm (x̃)
has full rank at x̃. Furthermore, the tangent space of V at x̃ is, of course, not contained in the
£
¤T
hyperplane x0 = 0 at infinity, so the column 1 0 . . . 0 is not in the column space of the
matrix at x̃. Thus already the submatrix
 ∂ ˜

∂ ˜
∂ ˜
∂x1 f1 (x̃)
∂x1 f2 (x̃) . . . ∂x1 fm (x̃)


..
..
..
..


.
.
.
.
∂ ˜
∂ ˜
∂ ˜
∂xn f1 (x̃) ∂xn f2 (x̃) . . . ∂xn fm (x̃)
has full rank, i.e., the gradients
∇f˜1 (x̃), . . . , ∇f˜m (x̃)
are linearly independent at x̃ ∈ V .
(ii) An optimizer x∗ must belong to V , and by (i), the gradients
∇f1 (x∗ ), ∇f2 (x∗ ), . . . , ∇fm (x∗ )
are linearly independent. Hence the KKT condition holds at x∗ (cf. [10, Chapter 12]).
(iii) We claim that the intersection of the complete intersection U and the variety W where
the Jacobian matrix M has rank at most m is finite. Since our critical points V ∩ W form a
subset of U ∩ W, (iii) would follow. The codimension of U is m, and this complete intersection
variety is smooth, so the matrix M has, by (i), rank at least m at each point of U. The variety
U ∩ {f˜0 (x̃) = 0} is, by Bertini’s Theorem A.1, also smooth. So as above, the matrix M has full
rank at the points in the affine part V ∩ {f0 (x) = 0}. On the other hand, M is the Jacobian
matrix of the variety U ∩ {f˜0 (x̃) = 0}. This variety is again smooth and has codimension m + 1
in the hyperplane {x0 = 0}, so M must have full rank m + 1 on U ∩ {f˜0 (x̃) = 0}. Therefore
the variety W, where M has rank at most m, cannot intersect U ∩ {f˜0 (x̃) = 0}. But Bézout’s
Theorem A.3 says that if the sum of the codimensions of two varieties in Pn does not exceed n,
then they must intersect. In particular, any curve in U intersects the hypersurface {f˜0 (x̃) = 0}.
Since U ∩ {f˜0 (x̃) = 0} has codimension m + 1, we deduce that W must have codimension at least
n − m. Furthermore, since any curve in U ∩ W would intersect {f˜0 (x̃) = 0}, the intersection
U ∩ W must be empty or finite. On the other hand, the variety of n × (m + 1)-matrices having
rank no more than m, whose entries are homogeneous polynomials in the coordinates of points
in a projective space, has codimension at most n − m, by Proposition A.5. So the codimension
of W equals n − m. Hence U and W have complementary dimensions. Therefore the intersection
U ∩ W is non-empty and (iii) follows.
¤
By Proposition 2.1, for generic polynomials f1 , . . . , fm , the optimal solutions of (1.1) can be
characterized by the KKT system (2.1), and for generic objective function f0 the KKT variety
K is finite. Geometrically, the algebraic degree of the optimization problem (1.1), under the
genericity assumption, is equal to the number of distinct complex solutions of KKT, i.e., the
cardinality of the variety K. We showed above that K coincides with V ∩ W. The variety
U ∩ W above clearly contains K. On the other hand, U ∩ W is finite and does not intersect
the hyperplane {x0 = 0} when the polynomials fi are generic. Since U\V = U ∩ {x0 = 0} and
6
JIAWANG NIE AND KRISTIAN RANESTAD
U ∩ W ∩ {x0 = 0} = ∅, we may conclude that K = U ∩ W. Therefore the cardinality of K must
coincide with the degree of the variety U ∩ W.
Let Sr be the r-th complete symmetric function on k letters (n1 , n2 , . . . , nk ):
X
(2.4)
Sr (n1 , n2 , . . . , nk ) =
ni11 · · · nikk .
i1 +i2 +···+ik =r
Theorem 2.2. Consider the polynomial optimization problem (1.1), and assume m = me , i.e.,
all contraints are active. If the polynomials f0 , f1 , . . . , fm are generic, then the algebraic degree
of (1.1) is equal to
d1 d2 · · · dm Sn−m (d0 − 1, d1 − 1, . . . , dm − 1).
Furthermore, if the polynomials f0 , f1 , . . . , fm are not generic, but the system (2.1) is still zerodimensional, then the above formula is an upper bound for the algebraic degree of (1.1).
Proof. When f1 , f2 , . . . , fm are generic, U is a smooth complete intersection of codimension m.
Its degree deg(U) = d1 d2 · · · dm by Corollary A.4. When f0 is also generic, W has codimension
n − m and intersects U in a finite set of points according to Proposition 2.1. If the intersection
U ∩ W is transverse
¡
¢ (i.e., smooth) and hence consists of a collection of simple points, then the
degree deg U ∩ W counts the number of intersection points of U ∩ W, and hence the cardinality
of the KKT variety K, which is also the number of complex solutions to the KKT system (2.1)
for problem (1.1).
To show that this intersection is transversal, we consider the subvariety X in Pn × Pm defined
by the m equations f˜1 (x̃) = f˜2 (x̃) = · · · = f˜m (x̃) = 0 and the n equations
M · (λ0 , . . . , λm )T = 0,
where the λi are homogeneous coordinate functions in the second factor. The image under the
projection of the variety X defined by these m + n polynomials into the first factor coincides
with the finite set U ∩ W. Since M has rank at least m at every point of U, there is a unique
λ̃ = (λ0 , . . . , λm ) ∈ Pm for each point x̃ ∈ U ∩ W such that (x̃, λ̃) lies in X. Therefore the variety
X is a complete intersection in Pn × Pm . When the coefficients of f0 vary, it is easy to check that
this complete intersection does not have any fixed point. This is because when the coefficients
of f0 vary, the common zeros of the n equations M · (λ0 , . . . , λm )T = 0 vary without fixed points.
So Bertini’s Theorem A.1 applies to conclude that for generic f0 this complete intersection is
transversal, which implies that the intersection U ∩ W in Pn is also transversal.
Since the intersection U ∩ W is finite, i.e., it has codimension in Pn equal to the sum of the
codimensions of U and W, Bézout’s Theorem A.3 applies to compute the degree
deg(U ∩ W) = deg(U) · deg(W).
To complete the computation, we therefore need to find deg(W). Since the codimension of W
equals the codimension of the variety defined by the (m + 1) × (m + 1) minors of a general
n × (m + 1) matrix with polynomial entries, the formula in Proposition A.6 applies to compute
this degree: The degree of W equals the degree of the determinantal variety of n × (m + 1)
matrices of rank at most m in the space of matrices whose entries in the i-th column are generic
forms of degree di − 1, i.e., Sn−m (d0 − 1, d1 − 1, . . . , dm − 1). Therefore the degree formula for
the critical locus U ∩ W and hence the algebraic degree of (1.1) is proved.
Assume that the polynomials fi are not generic, while the system (2.1) is still zero-dimensional.
Then a perturbation argument can be applied. Let x∗ be one fixed optimal solution of optimization problem (1.1). Apply a generic perturbation ∆² fi to each fi so that (fi + ∆² fi )(x) is a
generic polynomial and the coefficients of ∆² fi tends to zero as ² → 0. Then one optimal solution
ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION
7
x∗ (²) of the perturbed optimization problem (1.1) tends to x∗ . By genericity of (fi + ∆² fi )(x),
we know
a0 (²)(x∗i (²))δ + a1 (²)(x∗i (²))δ−1 + · · · + aδ−1 (²)x∗i (²) + aδ (²) = 0.
Here δ = d1 d2 · · · dm Sn−m (d0 − 1, d1 − 1, . . . , dm − 1) and aj (²) are rational functions of the
coefficients of fi and ∆² fi . Without loss of generality, we may normalize aj (²) such that
max |aj (²)| = 1.
0≤j≤δ
When ² → 0, by continuity, x∗i is a root of some univariate polynomial whose degree is at most
δ and coefficients are rational functions of the coefficients of polynomials f0 , f1 , . . . , fm .
¤
Remark 2.3. The genericity assumption in Theorem 2.2 is used to conclude that the critical
locus U ∩ W is a smooth zero-dimensional variety, i.e., a set of points, by appealing to Bertini’s
Theorem A.1, while Bézout’s Theorem A.3 counts its degree, i.e., the number of points. So both
theorems are needed to get the sharp degree bound. A sufficient condition for Bertini’s Theorem
to apply can be expressed in terms of the sets Ui of polynomials in which the polynomials
f0 , f1 , . . . , fm can be freely chosen. First, assume that the generic polynomial in each Ui is
reduced, and that Ui intersects every Zariski open set of a complex affine space Vi . Second,
assume that the set of common zeros of all the polynomials in ∪m
i=0 Vi is empty. Then Bertini’s
Theorem applies. In fact, the polynomials fi for which the conclusion of Bertini’s Theorem fails
are contained in a complex subvariety of Vi .
If some of the polynomials fi are reducible, then we may replace fi by the factor of least
degree that contains the optimizer. The original problem (1.1), is then modified to one with a
smaller algebraic degree. This is relevant in the above context, if the generic polynomial in Ui
is reducible.
Example 2.4. Consider the following special case of problem (1.1)
f0 (x) = 21x22 − 92x1 x23 − 70x22 x3 − 95x41 − 47x1 x33 + 51x22 x23 + 47x51 + 5x1 x42 + 33x53 ,
f1 (x) = 88x1 + 64x1 x2 − 22x1 x3 − 37x22 + 68x1 x22 x3 − 84x42 + 80x32 x3 + 23x22 x23 − 20x2 x33 − 7x43 ,
f2 (x) = 31 − 45x1 x2 + 24x1 x3 − 75x23 + 16x31 − 44x21 x3 − 70x1 x22 − 23x1 x2 x3 − 67x22 x3 − 97x2 x23 .
Here m = me = 2. By Theorem 2.2, the algebraic degree of the optimal solution is bounded by
4 · 3 · S1 (4, 3, 2) = 12 · (4 + 3 + 2) = 108.
A symbolic computation over the finite field Z/17Z, using Singular [5], shows the optimal coordinate x1 is a root of a univariate polynomial of degree 108. In this case the degree bound
108 is sharp. We were not able to find the exact coefficients of this univariate polynomial in the
rational field Q, since Singular could not complete the computation over Q.
Now we consider the more general case when m > me , i.e., there are inequality constraints.
Then a similar degree formula as in Theorem 2.2 can be obtained, as soon as the active set is
identified.
Corollary 2.5. Consider the polynomial optimization problem (1.1). Let x∗ be an optimizer
and let j1 , . . . , jk ⊂ {me + 1, . . . , m} be the indices of the active set of inequality constraints. If
every active fi is generic, then the algebraic degree of x∗ is
d1 · · · dme dj1 · · · djk Sn−me −k (d0 − 1, d1 − 1, . . . , dme − 1, dj1 − 1, . . . , djk − 1).
If at least one of polynomials fi is not generic and the system (2.1) is zero-dimensional, then
the above formula is an upper bound for the algebraic degree.
8
JIAWANG NIE AND KRISTIAN RANESTAD
Proof. Note that x∗ is also an optimal solution of the polynomial optimization problem


minn
f0 (x)

x∈R
s.t.
fi (x) = 0, i = 1, . . . , me  .
fi (x) = 0, i = j1 , . . . , jk 
Hence the conclusion follows from Theorem 2.2.
¤
3. Some special cases
In this section we derive the algebraic degrees of some special polynomial optimization problems. The simplest special case is that all the polynomials fi in (1.1) have degree one, i.e., (1.1)
becomes a linear programming problem of the form (1.2). If the objective c is generic, precisely
n constraints will be active. So the algebraic degree is S0 (0, 0, . . . , 0) = 1. This is consistent
with what we observed in the introduction. Now let us look at other special cases.
3.1. Unconstrained optimization
We consider the special case that the problem (1.1) has no constraints. It becomes an unconstrained optimization. The gradient of the objective vanishes at any optimal solution. By
Theorem 2.2, the algebraic degree is bounded by Sn (d0 −1) = (d0 −1)n , which is exactly Bézout’s
number for the gradient polynomial system
∇f0 (x) = 0.
Since f0 can be chosen freely among all polynomials of degree d0 , Remark 2.3 applies to show
that the degree bound above is sharp.
Example 3.1. Consider the minimization of f0 (x) given by
f0 = x41 + x42 + x43 + x44 + x31 + x32 + x33 + x34 − 13x21 − 30x1 x2 − 9x1 x3 + 5x1 x4 + 11x22
− 3x3 x2 − 3x23 − 20x3 x4 − 13x2 x4 − 9x24 + x1 − 2x2 + 12x3 − 13x4 .
For the above polynomial, the algebraic degree of the optimal solution is 34 = 81. A symbolic
computation over the rational field Q, using Singular [5], shows that the optimal coordinate x1
of x∗ is a root of a univariate polynomial of degree 81:
80
9671406556917033397649408x81
1 + 195845982777569926302400512x1 + · · ·
+38068577951137724978419521685033466020527544236947408128x1
−2957438647420262596596093352763852215662185651072180992.
The degree bound 81 is sharp for this problem.
3.2. Quadratically constrained quadratic programming
Consider the special case that all the polynomials f0 , f1 , . . . , fm are quadratic. Then (1.1)
becomes a quadratically constrained quadratic programming (QCQP) problem which has the
standard form
min
x∈Rn
s.t.
xT A0 x + bT0 x + c0
xT Ai x + bTi x + ci ≥ 0, i = 1, . . . , `.
ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION
9
Here Ai , bi , ci are matrices or vectors of appropriate dimensions. The objective and all the
constraints are all quadratic polynomials. At an optimal solution, suppose m ≤ ` constraints
are active. By Corollary 2.5, the algebraic degree is bounded by
µ ¶
X
n
m
m
m
(3.1)
2 · Sn−m (1, 1, . . . , 1 ) = 2 ·
1=2 ·
.
| {z }
m
i0 +i1 +i2 +···+im =n−m
m times
The polynomials f0 , f1 , . . . , fm can be chosen freely in the space of quadratic polynomials, so
Remark 2.3 applies to show that the degree bound above is sharp.
Example 3.2. Consider the polynomials
f0 = −20 − 27x21 + 89x1 x2 + 80x1 x3 − 45x1 x4 + 19x1 x5 + 42x1 − 13x22 + 31x2 x3 − 79x2 x4
+ 74x2 x5 − 9x2 + 56x23 − 77x3 x4 − 2x3 x5 + 35x3 + 40x24 − 13x4 x5 + 60x4 + 58x25 − 84x5 ,
f1 = 33 + 55x21 − 41x1 x2 + 33x1 x3 − 61x1 x4 + 96x1 x5 + 12x1 + 74x22 − 90x2 x3 − 57x2 x4
− 52x2 x5 + 51x2 + 15x23 + 81x3 x4 + 87x3 x5 + 75x3 − 10x24 + 58x4 x5 + 33x4 + 83x25 − 23x5 ,
f2 = 8 − 9x21 + 56x1 x2 − 24x1 x3 + 81x1 x4 + 85x1 x5 − 99x1 − 77x22 − 75x2 x3 + x2 x4 + 38x2 x5
+ 23x2 − 97x23 − 14x3 x4 − 73x3 x5 + 65x3 + 3x24 − 14x4 x5 + 16x4 + 9x25 − 10x5 ,
f3 = 9 + 90x21 − 94x1 x2 − 22x1 x3 − 24x1 x4 + 78x1 + 32x22 − 48x2 x3 − 6x2 x4 + 80x2 x5 − 18x2 − 63x23
+ 66x3 x4 − 13x3 x5 + 88x3 − 45x24 − 92x4 x5 − 69x4 − 43x25 + 32x5 .
For the above polynomials, the QCQP problem is nonconvex. We consider those local optimal
solutions for which all the three inequalities
are active. By Corollary 2.5, the algebraic degree
¡n¢
of this problem is bounded by 2m m
= 80. A symbolic computation over the finite field Z/17Z,
using Singular [5], shows that the optimal coordinate x1 is a root of a univariate polynomial of
degree 80. The algebraic degree of this problem is 80 and the bound given by the formula (3.1)
is sharp. For this example, we were also not able to find the exact coefficients of this univariate
polynomial in the rational field Q, since Singular could not complete the computation over Q.
3.3. Second order cone programming
The second order cone programming (SOCP) problem has the standard form
min
(3.2)
x∈Rn
s.t.
cT x
aTi x + bi − kCi x + di k2 ≥ 0, i = 1, . . . , `
where c, ai , bi , Ci , di are matrices or vectors of appropriate dimensions. Let x∗ be an optimizer.
Since SOCP is a convex program, the x∗ must also be a global solution. By removing the square
root in the constraint, SOCP becomes the polynomial optimization problem
min
x∈Rn
cT x
s.t. (aTi x + bi )2 − (Ci x + di )T (Ci x + di ) ≥ 0, i = 1, . . . , `.
Without loss of generality, assume that the constraints with indices 1, 2, . . . , m are active at x∗ .
The objective is linear but the constraints are all quadratic. As we can see, the Hessian of the
constraints have the special form ai aTi − CiT Ci . Let ri be the number of rows of Ci . When
ri = 1, the constraint aTi x + bi − kCi x + di k2 ≥ 0 is equivalent to two linear constraints
−(aTi x + bi ) ≤ Ci x + di ≤ aTi x + bi .
10
JIAWANG NIE AND KRISTIAN RANESTAD
Thus, when every ri = 1, the problem reduces to a linear programming problem and hence has
algebraic degree one, because in this situation the polynomial (aTi x+bi )2 −(Ci x+di )2 is reducible.
When ri ≥ 2 and ai , bi , Ci , di are generic, the polynomial (aTi x + bi )2 − (Ci x + di )T (Ci x + di ) is
quadratic of rank ri + 1 and hence irreducible. Without loss of generality, assume 1 = r1 = r2 =
· · · = rk < rk+1 ≤ · · · ≤ rm . Then the problem (3.2) is reduced to
min
x∈Rn
s.t.
cT x
aTi x + bi + σi (Ci x + di ) ≥ 0, i = 1, . . . , k
(aTi x + bi )2 − (Ci x + di )T (Ci x + di ) ≥ 0, i = k + 1, . . . , m
where the scalar σi is chosen such that aTi x∗ + bi + σi (Ci x∗ + di ) = 0. By Corollary 2.5, the
algebraic degree of SOCP in this modified form is bounded by
(3.3)
m−k
2
· Sn−m (0, . . . , 0, 1, . . . , 1 ) = 2
| {z }
m−k times
m−k
·
µ
X
1=2
ik+1 +ik+2 +···+im =n−m
m−k
¶
n−k−1
·
.
m−k−1
When k = m, we have already seen that the algebraic degree is one.
For the sharpness of degree bound (3.3), we apply Bertini’s Theorem A.1 following Remark 2.3.
For every i = k + 1, . . . , m, define the set Ui of polynomials as




X
Ui = (aTi x + bi )2 −
αj2 (Ci x + di )2j : α1 , . . . , αri ∈ R .


1≤j≤ri
Next, define complex affine spaces Vi as follows:


Vi =

(aTi x + bi )2 −
X
βj (Ci x + di )2j : β1 , . . . , βri ∈ C
1≤j≤ri



, i = k + 1, . . . , m.
Then every set Ui intersects any Zariski open subset of the affine space Vi . On the other hand
the set of common zeros of the linear polynomials
aTi x + bi + σi (Ci x + di ), i = 1, . . . , k
and all the polynomials in the union
(3.4)
Sm
i=k+1 Vi
is contained in the set
¾
k
m ½
T
\
©
ª \
n
T
n ai x + bi = 0
Z=
x ∈ R : ai x + bi + σi (Ci x + di ) = 0
x∈R :
.
C i x + di = 0
i=1
i=k+1
Therefore, for generic choices ai , bi , Ci , di , if rk+1 + · · · + rm + m > n, the set Z is empty. Hence
Remark 2.3 applies to show that, for¡ generic
¢ choices of c, ai , bi , Ci , di with rk+1 +· · ·+rm +m > n,
n−k−1
the algebraic degree bound 2m−k · m−k−1
is sharp.
ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION
11
Example 3.3. Consider the SOCP defined by the polynomials
f0 = −x1 + 6x2 + 13x3 + 11x4 + 8x5 ,
f1 = (−11x1 − 18x2 − 4x3 + 2x4 − 12x5 + 7)2 − (−4x1 − 10x2 + 20x3 − 4x4 − 9x5 + 3)2
− (−5x1 − 11x2 + 8x3 − 18x4 + 11x5 + 15)2 − (21x1 + 18x2 − 12x3 − 10x4 − 8x5 + 4)2 ,
f2 = (−5x1 − 5x2 − 7x3 − 6x4 + 4x5 + 41)2 − (x1 − 2x2 + 10x3 − 21x4 − 11)2
− (−12x1 + 3x2 + 16x3 + 4x4 + x5 + 9)2 − (14x1 + 20x2 − 13x3 − 7x4 + 4x5 + 2)2 ,
f3 = (x1 − 8x2 + 11x3 − x5 + 22)2 − (2x1 − x2 + 3x3 − x4 − 25x5 − 8)2
− (−2x1 − 17x3 + 14x4 + 4x5 − 7)2 − (x1 + 12x2 + 14x3 − 6x4 − 4x5 − 10)2 .
There are no linear constraints. For this SOCP, all the three inequalities are active at the
optimizer. All the matrices
C¢i have three rows. By formula (3.3), the algebraic degree of this
¡5−1
3
problem is bounded by 2 3−1 = 48. A symbolic computation over the finite field Z/17Z, using
Singular [5], shows that the optimal coordinate x1 is a root of a univariate polynomial of degree
48. The algebraic degree of this problem is 48, so the upper bound is sharp in this case. The
exact integer coefficients of this polynomial are huge, e.g., the coefficient of x48
1 returned by
Singular is
2099375102740465860059815913466313028033389427217933637192605381170459911
3664919113955945081518362866289941284683755539037514999015805213743887244
0868003318410425145082276365726847061266590451302699523333919731587145180
8453244977937102564917173654129343733244846958387649410769452695126458577
1066157333966857752253305920226530568083266479375648347403229514209064223
5487138449138079371730302676639572182280037694111467584821502831737932897
4452926895048780181141913600.
3.4. p-th order cone programming
The p-th order cone programming (POCP) problem has the standard form
(3.5)
minx∈Rn cT x
s.t. aTi x + bi − kCi x + di kp ≥ 0, i = 1, . . . , `
where c, ai , bi , Ci , di are matrices or vectors of appropriate dimensions. This is also a convex
optimization problem. Let x∗ be an optimizer, and assume the constraints with indices 1, . . . , m
are active at x∗ . Suppose Ci has ri rows. When some ri = 1, the constraint aTi x + bi − kCi x +
di kp ≥ 0 is equivalent to two linear constraints
−(aTi x + bi ) ≤ Ci x + di ≤ aTi x + bi .
Like the SOCP case, assume 1 = r1 = · · · = rk < rk+1 ≤ · · · ≤ rm . Then the problem (3.5) is
equivalent to
min
x∈Rn
s.t.
cT x
aTi x + bi + σi (Ci x + di ) ≥ 0, i = 1, . . . , k
(aTi x
ri
X
(Ci x + di )pj ≥ 0, i = k + 1, . . . , m
+ bi ) −
p
j=1
12
JIAWANG NIE AND KRISTIAN RANESTAD
where the scalar σi is chosen such that aTi x∗ + bi + σi (Ci x∗ + di ) = 0. Then we have
X
¶
n−k−1
.
m−k−1
µ
ik+1 +···+im
(p − 1)
Sn−m (0, . . . , 0, p − 1, . . . , p − 1) =
|
{z
}
ik+1 +···+im =n−m
m−k times
n−m
= (p − 1)
By Corollary 2.5, the algebraic degree of x∗ is therefore bounded by
µ
¶
n−k−1
(3.6)
pm−k (p − 1)n−m
.
m−k−1
When k = m, problem (3.5) is reducible to a linear programming problem and hence its algebraic
degree is one.
Now we discuss the sharpness of degree bound (3.6). Similarly to the SOCP case, for every
i = k + 1, . . . , m, define the set of polynomials Ui as




X p
αj (Ci x + di )pj : α1 , . . . , αri ∈ R .
Ui = (aTi x + bi )p −


1≤j≤ri
Then define complex affine spaces Vi as follows:




X
βj (Ci x + di )pj : β1 , . . . , βri ∈ C , i = k + 1, . . . , m.
Vi = (aTi x + bi )p −


1≤j≤ri
Then every set Ui intersects any Zariski open subset of the affine space Vi . On the other hand,
the set of common zeros of the linear polynomials
aTi x + bi + σi (Ci x + di ), i = 1, . . . , k
S
and all the polynomials in the union m
i=k+1 Vi is contained in the set Z defined by (3.4).
Therefore, for generic choices of ai , bi , Ci , di with rk+1 + · · · + rm + m > n, the set Z is empty,
and hence Remark 2.3 implies that the degree bound given by formula (3.6) is sharp.
Example 3.4. Consider the case p = 4 and the polynomials
f0 = 9x1 − 5x2 + 3x3 + 2x4
f1 = (1 − 6x1 − 6x2 + 4x3 − 9x4 )4 − (7 − 6x1 + 22x2 − x3 + x4 )4
− (11 + x1 − x2 − 8x3 + 3x4 )4 − (−13 + 7x1 + 16x2 − 7x3 + 9x4 )4
− (3 − 11x1 + 14x2 − 8x3 + 5x4 )4 − (8 + 9x1 − 10x2 + 2x3 + 2x4 )4 .
For the above polynomials, the inequality constraint must be active since the objective is
linear. By the
formula
(3.6), the algebraic degree of the optimal solution is bounded by
¡ n−1
¢
m
n−m
p (p − 1)
m−1 = 108. A symbolic computation over the finite field Z/17Z, using Singular [5], shows that the optimal coordinate x1 is a root of a univariate polynomial of degree
108. So the algebraic degree of this problem is 108, and the bound given by the formula (3.6)
is sharp. For this example, we were also not able to find the exact coefficients of this univariate
polynomial in the rational field Q, since Singular could not complete the computation over Q.
3.5. Semidefinite programming
In the introduction, we observed that the SDP problem of the form (1.3) can also be represented as a polynomial optimization problem of the form (1.1). Concretely, let gI (x) be the
ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION
13
principle minor of the matrix in (1.3) with rows I ⊆ {1, 2, . . . , N } (N is the length of the matrix).
Then the problem (1.3) is equivalent to
(
min
cT x
x∈Rn
(3.7)
s.t.
gI (x) ≥ 0, ∀ I ⊆ {1, 2, . . . , N }.
If the active constraints are known in the above, then Corollary 2.5 can be applied to get
an upper bound for the algebraic degree of problem (1.3). Unfortunately, the upper bound
obtained by Corollary 2.5 is usually not sharp, and often larger than the degree formula given in
[13, Theorem 1.1]. This is because the polynomials gI (x) do not define a complete intersection:
The codimension of Vr is less than the number of minors, i.e., the number of generators of the
ideal of Vr . To see this point, suppose r is the rank of the optimal matrix in (1.3). Then all
gI (x) with card(I) > r must vanish at the optimal solution x∗ . Let Vr be the variety defined by
the gI (x):
Vr = {x ∈ Rn : gI (x) = 0, ∀ I : card(I) > r}
¡N ¢
The ideal I(Vr ) of Vr has r+1
generators. Since Vr contains the variety of matrices of rank
¡
¢
at most r, which has codimension N −r+1
, the codimension of Vr is smaller than the number
2
of generators of its ideal for almost all values of N and r. So the variety Vr is almost never a
complete intersection.
Acknowledgement. The authors are grateful to Gabor Pataki, Richard Rimanyi and Bernd
Sturmfels for motivating this paper, and wish to thank two referees for fruitful suggestions.
Appendix A. Some elements of algebraic geometry
This section presents some basic definitions and theorems in algebraic geometry. Most of
them can be found in Harris’ book [4].
Let C[x1 , . . . , xn ] be the ring of polynomials in x1 , . . . , xn . An additive subgroup I of C[x1 , . . . , xn ]
is called an ideal if for every f ∈ I, the product f · g ∈ I for any g ∈ C[x1 , . . . , xn ]. Given an
ideal I of C[x1 , . . . , xn ], the polynomials f1 , . . . , fk are called generators of I if for every f ∈ I
there exist g1 , . . . , gk ∈ C[x1 , . . . , xn ] such that f = f1 g1 + . . . + fk gk . We also say that the ideal
I is generated by f1 , . . . , fk and denote I = hf1 , . . . , fk i.
Let Cn be the n-dimensional complex affine space. A point x ∈ Cn is a complex vector
(x1 , . . . , xn ). A set V in Cn is called an affine algebraic variety if there are polynomials g1 , . . . , gr
such that
V = {(x1 , . . . , xn ) ∈ Cn : g1 (x1 , . . . , xn ) = · · · = gr (x1 , . . . , xn ) = 0} .
In the Zariski topology on Cn the closed sets are precisely the affine algebraic varieties. A variety
is irreducible if it is not the union of two proper closed subvarieties. Any variety is a finite union
of distinct irreducible varieties. Given an affine variety V , the ideal of V is defined to be the set
of polynomials
I(V ) = {f ∈ C[x1 , . . . , xn ] : f (x1 , . . . , xn ) = 0, ∀ (x1 , . . . , xn ) ∈ V } .
Pn
Let
be the n-dimensional complex projective space of lines through the origin in Cn+1 .
A point x̃ ∈ Pn has homogeneous coordinates [x0 , x1 , . . . , xn ], unique up to multiplication by
a common nonzero scalar. A set U in Pn is called a projective algebraic variety if there are
homogeneous polynomials h1 (x0 , x1 , . . . , xn ), . . . , ht (x0 , x1 , . . . , xn ) such that
U = {(x0 , x1 , . . . , xn ) ∈ Pn : h1 (x0 , x1 , . . . , xn ) = · · · = ht (x0 , x1 , . . . , xn ) = 0} .
14
JIAWANG NIE AND KRISTIAN RANESTAD
In the Zariski topology on Pn , the closed sets are precisely the projective algebraic varieties. As
a special case, if h ∈ C[x0 , . . . , xn ] is a homogeneous polynomial, then the set
H = {x̃ ∈ Pn : h(x̃) = 0} ⊂ Pn
is a projective variety called a hypersurface. If furthermore h has degree one, H is called a
hyperplane.
A subset I of C[x0 , . . . , xn ] is called a homogeneous ideal if I is an ideal of C[x0 , . . . , xn ] and
generated by homogeneous polynomials in the ring C[x0 , . . . , xn ]. Given a projective variety U,
the ideal
I(U) = {f ∈ C[x0 , x1 , . . . , xn ] : f (x̃) = 0, ∀ x̃ = (x0 , x1 , . . . , xn ) ∈ U}
is homogeneous. It is called the ideal of U. A Zariski open subset Q of a projective variety U
is called a quasi-projective variety, or equivalently, a quasi-projective variety is a locally closed
subset of Pn in the Zariski topology.
The dimension of an affine (resp. projective) variety V is the length k of the longest chain
of irreducible affine (resp. projective) subvarieties, V = V0 ⊃ V1 ⊃ · · · ⊃ Vk . For a projective
variety the dimension may be equivalently defined as the largest integer k such that any set
of k hyperplanes have a common intersection point on V . A variety has pure dimension if
all its irreducible components have the same dimension. Let U be a projective variety in Pn
of pure dimension k. We say that U is a complete intersection if its homogeneous ideal is
generated by n − k homogeneous polynomials, that is, there exists homogeneous polynomials
f1 , . . . , fn−k ∈ C[x0 , x1 , . . . , xn ] such that I(U) = hf1 , . . . , fn−k i.
Let U be an irreducible projective variety in Pn of dimension k and I(U) = hf1 , . . . , fr i. The
singular locus Using is defined to be the projective algebraic variety
Using = {x̃ ∈ U : J(f1 , . . . , fr ) has rank less than n − k at x̃} ,
where J(f1 , . . . , fr ) denotes the Jacobian matrix of f1 , . . . , fr .
The following is a fundamental theorem that we formulate in a version particularly applicable
to our situation. We consider subvarieties in products of projective spaces. They are defined, as
above, by ideals of polynomials that are homogeneous in two sets of variables, one set for each
factor.
Theorem A.1 (Bertini’s Theorem). If X ⊂ Pn1 × Pn2 is a quasi-projective k-dimensional
complex variety, and Pm a projective space parameterizing hypersurfaces in Pn1 × Pn2 . Let
Z = ∩H∈Pm H be the common points of these hypersurfaces. Let Y = H ∩ X for a general
member H ∈ Pm . Then Y is a variety of dimension k − 1 and
Ysing ⊂ (Xsing ∩ Y) ∪ Z.
Proof. Let fH be a polynomial that defines H. The polynomials {fH : H ∈ Pm } generate a
vector space V of dimension m + 1 of polynomials. Let f0 , . . . , fm be a basis for this vector
space. Then
x 7→ [f0 (x), . . . , fm (x)]
defines a regular map
X \ Z → Pm .
The conclusion now follows from [4, Theorems 17.16 and 17.24].
An immediate corollary is:
¤
ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION
15
Corollary A.2. Let f1 , . . . , fk ∈ C[x0 , . . . , xn1 , y0 , . . . , yn2 ] be generic polynomials and k ≤ n.
Then the projective variety V (f1 , . . . , fk ) defined by
n
o
(x̃, ỹ) = ([x0 , x1 , . . . , xn1 ], [y0 , . . . , yn2 ]) ∈ Pn1 × Pn2 : f1 (x̃, ỹ) = · · · = fk (x̃, ỹ) = 0
is a smooth complete intersection of dimension (n1 + n2 − k).
The degree of an equidimensional projective variety X of dimension k in Pn is the number
of points in the intersection of X with n − k general hyperplanes. Therefore if X and Y are
subvarieties of Pn of the same dimension without common components, then
deg(X ∪ Y) = deg(X ) + deg(Y).
The following is another fundamental theorem, which we also formulate in a version particularly
applicable to our situation.
Theorem A.3 (Bézout’s Theorem). If X , Y ⊂ Pn are projective complex varieties of pure
dimensions k and ` with k + ` ≥ n, then X ∩ Y =
6 ∅ and each irreducible component of the
intersection has dimension at least k + ` − n.
If X and Y intersect transversely, i.e., that each irreducible component of X ∩Y has dimension
k + ` − n and has a nonempty smooth open subset outside the singular loci of X and Y, then
deg(X ∩ Y) = deg(X ) · deg(Y).
In particular, if k + ` = n and X ∩ Y is transverse, then it consists of deg(X ) · deg(Y) points.
Proof. The first part is a secial case of [4, Theorem 17.24], except for the statement that the
intersection X ∩ Y =
6 ∅, while the second part is [4, Theorem 18.3]. To show that the intersection
X ∩Y =
6 ∅ we pass to the affine cones CX and CY in Cn+1 of the projective varieties in X and
Y respectively. They, of course, both contain 0, so their intersection in Cn+1 is non-empty. The
cones CX and CY have dimensions k+1 and `+1, respectively. Therefore, by [4, Theorem 17.24],
their intersection must have components of dimension at least (k + 1) + (` + 1) − n + 1 ≥ 1.
The intersection CX ∩ CY is clearly an affine cone over the intersection X ∩ Y, so since it has
dimension at least one, the intersection X ∩ Y is not empty.
¤
Corollary A.4. A complete intersection of hypersurfaces has degree equal to the product of the
degrees of the hypersurfaces.
Bertini’s Theorem and Bézout’s Theorem may be generalized to projective algebraic varieties
whose ideal is given by all minors of a given size of a matrix whose entries are homogeneous
polynomials. The generalization of Bertini’s Theorem that we need is given by the following:
Proposition A.5. Let a1 ≤ · · · ≤ af be a finite sequence of positive integers, and let M be
an e × f -matrix whose entries mi,j are homogeneous polynomials in C[x0 , . . . , xn ] of degree ai .
Consider the projective variety Xr = {x̃ ∈ Pn : M has rank at most r at x̃}. Every irreducible
component of Xr has dimension ≥ (n − (f − r)(e − r)). In particular, if n − (f − r)(e − r) ≥ 0,
then Xr 6= ∅. Furthermore, if the entries in M are generic, then every irreducible component of
Xr has dimension n − (m − r)(k − r), and Xr−1 is the singular locus of Xr .
Proof. The bound on the codimension follows from [4, Proposition 17.25]. For the smoothness
we consider first the case when the entries are independant variables y0 , . . . , yef −1 . A local
parameterization of Xr then shows that Xr has codimension (e−r)(f −r), is smooth outside Xr−1
and singular along Xr−1 , (cf. [4, Example 20.5]). Substituting the variables with polynomials
in x0 , . . . , xn , we may apply Bertini’s Theorem A.1 to conclude.
¤
16
JIAWANG NIE AND KRISTIAN RANESTAD
To find the degree of Xr , we apply Bezout’s Theorem in a slight variation of an argument in
[4, Example 19.10] to get the special case r = min{e − 1, f − 1} that we need. For general r the
formula is given by the Thom-Porteous Formula, whose proof is more involved (cf. [3, Example
14.4.11]).
Proposition A.6. Let a1 ≤ · · · ≤ af be a finite sequence of positive integers, and let M be an
e × f -matrix whose entries mi,j are general homogeneous polynomials in C[x0 , . . . , xn ] of degree
aj . Consider the projective variety Xe−1 = {x̃ ∈ Pn : M has rank less than e at x̃}.
(1) Assume that e ≤ f and let E1 , . . . , Ef be the elementary symmetric polynomials in the
ai , i.e.,
(1 + a1 t) · · · (1 + af t) = 1 + E1 t + · · · + Ef tf .
If n − f + e − 1 ≥ 0, then
degXe−1 = Ef −e+1 .
(2) Assume that e > f and let S1 , . . . , Sk , . . . be the complete symmetric polynomials in the
ai , i.e.,
1
=
(1 − a1 t)(1 − a2 t) · · · (1 − af t)
(1 + a1 t + a21 t2 + · · · ) · · · (1 + af t + a2f t2 + · · · ) = 1 + S1 t + · · · + Sk tk + · · ·
If n − e + f − 1 ≥ 0, then
degXf −1 = Se−f +1 .
Proof. The first part we prove by specializing the matrix to one where the entries of each column
in M are proportional, while any f − e + 1 polynomials from distinct columns define a complete
intersection. Clearly then Xe−1 is simply the union of these complete intersections; M has rank
less than e precisely when at least f − e + 1 of the columns vanish. The degree of each of these
complete intersections is the product of degrees of the corresponding polynomials, and their
union have a degree equal to the sum Ef −e+1 .
We prove the second formula by induction. Assume that n > 1 and that the formulas hold
when e < n, the case e = 1 being trivial. Assume that e = n > f , and let Mi be the submatrix
of M consisting of the first e − i rows, and Nj the submatrix of M consisting of the rows
e − f, e − f + 1, . . . , e − j. Let Xi be the variety of points where Mi has rank less than f , while
Yj is the variety of points where Nj has rank less than f − j. By induction deg Xi = Se−i−f +1 ,
while degYj = Ej+1 by the previous argument. Notice that X ⊂ X1 ∩ Y0 and that
Xf ∩ Yf −1 ⊂ · · · ⊂ Xi ∩ Yi−1 ⊂ · · · ⊂ X1 ∩ Y0 .
Furthermore
X = (X1 ∩ Y0 \ (X2 ∩ Y1 \ (X3 ∩ Y2 · · · \ (Xf ∩ Yf −1 )) · · · ).
So by induction
degX = Se−f E1 − Se−f −1 E2 + · · · + (−1)f −1 Se−2f +1 Ef .
Computing the coefficient of te−f +1 in the identity
(1 + a1 t) · · · (1 + af t)
= (1 + E1 t + · · · + Ef tf )(1 − S1 t + · · · + (−1)k Sk tk + · · · ),
1=
(1 + a1 t) · · · (1 + af t)
we get
Se−f E1 − Se−f −1 E2 + · · · + (−1)f −1 Se−2f +1 Ef = Se−f +1 .
So the second part of the proposition also holds.
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ALGEBRAIC DEGREE OF POLYNOMIAL OPTIMIZATION
17
References
[1] F. Catanese, S. Hoşten, A. Khetan and B. Sturmfels: The maximum likelihood degree, American Journal of
Mathematics, 128 (2006) 671–697.
[2] D. Cox, J. Little and D. O’Shea. Ideals, varieties, and algorithms. An introduction to computational algebraic
geometry and commutative algebra. Third edition. Undergraduate Texts in Mathematics. Springer, New York,
2007.
[3] W. Fulton. Intersection Theory. Springer Verlag, 1984.
[4] J. Harris. Algebraic Geometry, A First Course. Springer Verlag, 1992.
[5] G.-M. Greuel, G. Pfister and H. Schoenemann. SINGULAR: A Computer Algebra System for Polynomial
Computations. Department of Mathematics and Centre for Computer Algebra, University of Kaiserslautern.
http://www.singular.uni-kl.de/index.html
[6] S. Hosten, A. Khetan and B. Sturmfels. Solving the likelihood equations. Foundations of Computational
Mathematics. 5 (2005) 389-407.
[7] J. B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM J. Optim., 11(3):
796-817, 2001.
[8] J. Nie, J. Demmel and B. Sturmfels. Minimizing polynomials via sum of squares over the gradient ideal.
Math. Prog., Series A, Vol. 106 (2006), No. 3, pp. 587-606.
[9] J. Nie, K. Ranestad and B. Sturmfels. The algebraic degree of semidefinite programming. To appear in
Mathematical Programming.
[10] J. Nocedal and S. Wright. Numerical Optimization, Springer Series in Operations Research, Springer-Verlag,
New York, 1999.
[11] P. Parrilo. Semidefinite programming relaxations for semialgebraic problems. Math. Prog., Ser. B, Vol. 96,
No.2, pp. 293-320, 2003.
[12] P. A. Parrilo and B. Sturmfels. Minimizing polynomial functions. In S. Basu and L. Gonzalez-Vega, editors,
Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science,
volume 60 of DIMACS Series in Discrete Mathematics and Computer Science, pages 83-99. AMS, 2003.
[13] K. Ranestad and H.C. Graf von Bothmer. A general formula for the algebraic degree in semidefinite programming. 6 pages. To appear in Bulletin of LMS, available at http://arxiv.org/abs/math/0701877
Department of Mathematics, UC San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
E-mail address: njw@math.ucsd.edu
Department of Mathematics, University of Oslo, PB 1053 Blindern, 0316 Oslo, Norway
E-mail address: ranestad@math.uio.no